Expert: Daniel Mazur - 8/28/2006

Question
Hey Daniel,

You always serve my queries better than I expect, thanks a lot again. Well, my course structure is available here : http://www.nottingham.ac.uk/physics/courses/ug_courses_list.php?code=000283&page ; This new uni i have got into is highly reputed in UK, for physics and medicine in particular. So I guess whatever they teach must be upto international standards. (Plus a degree from a good uni might raise my chances of going to US for further studies, something i've always wanted to do :)! )

Yeah, I do know about matrices, vector calculus, double-triple integration, line integrals etc but at a very basic level .... a level from which i can't see how it connects with quantum or maybe particle physics eventually. But I'm working on it. Reading mostly from books i got from internet.

as far as matlab, maple goes ... perhaps you are right. Should wait for classes. As a physicist enthusiast, you know .. it happens. I wish to know everything that i can, but dunno where to start with. there's so much to learn.

Can you suggest me some mathematics books and very specific topics that i should start working on right now ? And some good techniques for inducing physics problems in all the math i learn...

waiting for your reply.

yours sincerely,
Bob.
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Followup To

Question -
Hey Daniel,
not sure if you remember me or not. i asked you long back about the programming side of physics and career in particle physics and stuff. well, thanks again for your advice. was helpful.
i just moved to university of nottingham from a rather unknown nottingham trent uni. got admission based on my good grades in first year.
at uni nottingham, we do have lots of work on computers. but its quite different from what i expected. i am asked to work on Matlab and Maple software in vacations. I've just started learning them but questions pop up in mind anyhow. i check internet but as always internet finds everything except what is really required. i can hardly find any basic info. thought why not just ask the expert! although i'm learning from tutorial files that matlab provides and linear algaebra, its hardly making any sense to me right now. [ Guess i should wait for classes and use notes for the time-being but i'm not a patient man :) ]
do you know where this leads to? How should i go about matlab? and what is mathematical modelling?
and last but not least, what role does it play in study of particle physics if any?

yours sincerely,
Bob.

Answer -
Hello again, Bob,

I am glad you valued my avice before and I am ready to answer these new question.

Mathematical modeling is a priceless tool in all branches of science, engineering and a few others like stock markets trading. Having a theory describing a nature's law is an excellent thing by itself, but to make a real use of it, you need to work with it actively. By this I mean that you take your problem (say "What is the dependence of the outcome of a proton-proton collision on energy?"), calculate, what existing theories predict, and finally prepare your experiment, select detectors and data acquisition system accordingly. And of course, after you have the data, you compare it with the model and if agreement is not good enough, you try to expand your models... The "calculation of theoretical predictions" is called mathematical modelling. That's what you do for every physics labs, however simple the problem is. When your problem is more complex, involves iterative solutions of equations that cannot be solved analyticaly, then you would call it "computer modelling".

Mathematical (computer) modelling is extemely useful, although it has its downfalls. A well done modelling can save you lots of time and money. Aviation industry is a very good example, every airplane that explodes in your computer model saves you all the money that a real airplane costs, see? But of course, if you overlook some limitation of your model's validity, your may be misleaded into building something that cannot fly either. However, yes, particle physics takes a good advantage of the modelling and it is an essential skill, mainly through probability and statistical theory.

All the maths you are learning at this early stage has a lot to do with the physics you will encounter later. Please do pay the Linear Algebra all attention it requires, even though it probably won't seem very useful for some time yet. Elementary particles have quantum character and at least half of the maths needed for the Quantum Theory is the  Linear Algebra (vectors, matices, linear operators,...). I thought it best to tackle it as it came in class, rather than getting ahead of the game. Something similar applies to Matlab and Maple. It is virtually impossible to apreciate the tools these packages give you to deal with problems even as simple as systems of linear equations, until you are at least introduced to what matrix is and the "legal" operations with them... Of course, you may already know all about matrices, do you? Please send me a follow-up question with details of what you are doing right now and we can discuss things further.

May what I have written serve you well!
Daniel

Answer
Hello Bob,

once you know about matrices and vectors, you can start using Matlab efficiently (I cannot speak for Maple, I never really used it). This is because Matlab is optimized for handling of arrays and parallel computing wherever possible. Apart from the general advice to programming using the internal array operations (rather than FOR-loops), scientific programing in Matlab is just like in any other language, C or Fortran. You can use it to calculate the same simulations, like the three-body problem, or non-linear equations like rocket accelerations in a gravitational field...

It's perhaps better to say you can program a simulation to ANY physical problem. It's only up to you to choose the model and you're ready to go. My point of saying "tackle the problems as they come" was, that one first has to really understand the physics of what is going on, and only then one can confidently program the simulation.

Let's say that at your stage you really understand Newtonean mechanics, or relativistic mechanics, so you probably ought to focus on learning Matlab trying to program simulations in Mechanics. You will need a textbook on Mathematical Physics and programing methods to learn the best algorithms, so you don't spend ages inventing them yourself. This is unfortunately put off until late (F3CAM7), so you better get a little ahead in this. I was learning mathematical methods for numerical simulations in my language (Czech) and we studied only from texts prepared by our teacher, so I cannot honestly recommend a textbook on this. However, just a quick search in Google returned me a link to Wikipedia, where you can find some of those useful algorithms: Euler integration, Verlet integration, Runge-Kutta algorithm... All these come in as methods of solving ordinary differential equations. Any basic textbook on Computational Science should do, you can look through indexes of many of them on Amazon, so I am confident you will find an appropriate one easily. Maybe, asking your Modelling professor is the best way to proceed anyway.

Back to the Newtonean mechanics: I am sure you have learned the basics of ODRs already, so you can sit down and with the aid of those algorithms above you can program a simulation of the planetary movements in Solar System. I am sure you will have learned A LOT, by the time you have your model running: about Matlab, about the algorithms and about the challenges of computer simulations in general. Good luck!

As far as the connection to particle physics goes, there is a long way, agreed. The Solar System is a classical system, not quantum system. But I strongly suggest you learn modelling on a system of classical particles. This is because you can compare the result of the model with your everyday experience, so you easily detect errors you make in the code. Quantum systems behave oddly (from the point of view of Classical Physics) and you cannot use the "common sense" as feedback there.

Now about the "techniques for inducing physics problems in all the math"... this is a serious teacher's job. This is what every teacher should do for you, to keep your interest in the subject and at the same time it is rather difficult even for the teachers themselves. If you are taught Maths by mathematicians, they won't necessarily have the strong physics background to tell you real-life examples right away. Moreover, it is really difficult for the basic maths we learn in the first and second year, those are the basics needed for EVERYTHING, but you need to know more (another year of Maths) to make it all really useful. In a mechanistic picture you can consider the first year of maths as an artist learning about the properties of clay and wood and metal, while really longing to start creating sculptures and carvings. The thing missing in the middle is how to use the furnace and the chisel and hammer to make the materials do what you want. That's how I would picture it:-)))

Now, the best way to find applications of your maths load would be to read some advanced physics textbook first. It will point you to the prerequisites and going down this chain you'll find the maths underlying it all.
Example: Particle Physics
-> Quantum mechanics
----> Linear Algebra (operators,Eignvalues,eigenvectors), Partial Diff.Eqns (Wave eqn), Theory of Measure, total and partial derivatives,...
-> Statistical physics
----> Integration, Sums of series,...

Well, I hope I gave you the picture:-). Incidentally, when learning Matlab, you should use their discussion forum. I sometimes post there some basic questions and I am already used to getting replies like "read the Getting Started first";-) One must not be frightened off.

Take care and write again, if needed.

Daniel